Problem: All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$3.00$ each for students, and the group paid $$37.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$13.00$ each for teachers and $$12.50$ each for students, and the group paid $$114.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+3y = 37.5}$ ${13x+12.5y = 114}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-13x-6y = -75}$ ${13x+12.5y = 114}$ Add the top and bottom equations together. $ 6.5y = 39 $ $ y = \dfrac{39}{6.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {6.5x+3y = 37.5}$ to find $x$ ${6.5x + 3}{(6)}{= 37.5}$ $6.5x+18 = 37.5$ $6.5x = 19.5$ $x = \dfrac{19.5}{6.5}$ ${x = 3}$ You can also plug ${y = 6}$ into $ {13x+12.5y = 114}$ and get the same answer for $x$ ${13x + 12.5}{(6)}{= 114}$ ${x = 3}$ There were $3$ teachers and $6$ students on the field trips.